Finite Group Databasebeta

← #28335415

Subgroups — #28335415

Order 13696.

308 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293189C211yesmaximal, characteristic
2#20292768C212yesmaximal, complemented
3#20292539C211yesmaximal, characteristic, complemented
4#20292362C212yesmaximal, complemented
5#323×C107C211yesmaximal, characteristic, complemented
6C22×Dic214C2211yescharacteristic, complemented
7C22×C22⋊D107—21no—
8C2×#13133807—22no—
9#16812093—22no—
10#16812083—21no—
11#16812083C2212yes—
12#16812079—22no—
13C2×C22⋊Dic107—21no—
14#16812034—22no—
15C107⋊#95—21no—
16C107⋊#95C2211yescharacteristic, complemented
17C2×C22⋊C4×C107C2211yescharacteristic
18C22⋊C8×C107C2212yescomplemented
19C23×D107—21no—
20C2×Dic214—22no—
21C2×Dic214—42no—
22C2×D428—21no—
23C2×D428D412yes—
24C2×C22⋊D107—22no—
25C2×C22⋊D107—22no—
26C2×C22⋊D107—42no—
27C2×C22⋊D107—42no—
28C22×Dic107—21no—
29C22×Dic107—21no—
30C22×Dic107—21no—
31#13133807—44no—
32C4⋊Dic107—21no—
33C4⋊Dic107D412yes—
34C22⋊Dic107—42no—
35(C2×D107)⋊C4—22no—
36(C2×D107)⋊C4—22no—
37C24×C107D411yescharacteristic
38C22×C4×C107D411yescharacteristic
39C22×C4×C107C2311yescharacteristic
40C2×C8×C107—22no—
41C22⋊C4×C107—22no—
42C23×C107—21no—
43C23×C107—21no—
44C23×C107—41no—
45C23×C107—41no—
46C23×C107SD1612yescomplemented
47C23×C107C2×D411yescharacteristic
48C22×D107—22no—
49C22×D107—42no—
50C22×D107—81no—
51C22×D107—81no—
52C2×C4×C107—21no—
53C2×C4×C107—41no—
54C2×C4×C107—41no—
55C2×C4×C107C2×D412yes—
56C22⋊D107—44no—
57C22⋊D107—42no—
58C22⋊D107—42no—
59C2×Dic107—22no—
60C2×Dic107—22no—
61C2×Dic107—42no—
62C2×Dic107—41no—
63C2×Dic107—41no—
64C2×Dic107—41no—
65C2×Dic107—41no—
66C2×Dic107—41no—
67D428—42no—
68D428—81no—
69Dic214—24no—
70Dic214—42no—
71C8×C107—42no—
72C22×C107—21no—
73C22×C107—21no—
74C22×C107—22no—
75C22×C107—42no—
76C22×C107—41no—
77C22×C107—41no—
78C22×C107—81no—
79C22×C107#12011yescharacteristic
80C22×C107C2×SD1611yescharacteristic
81C22×C107AGL(1,ℤ/8)11yescharacteristic
82C2×D107—42no—
83C2×D107—42no—
84C2×D107—42no—
85C2×D107—84no—
86C4×C107—22no—
87C4×C107—41no—
88Dic107—42no—
89Dic107—42no—
90Dic107—81no—
91C2×C107—21no—
92C2×C107—21no—
93C2×C107—42no—
94C2×C107#45011yescharacteristic
95C2×C107#45511yescharacteristic
96C2×C107B(4,𝔽2)11yescharacteristic
97D107—82no—
98#1601—1071nomaximal, Sylow
99C107#160111yescharacteristic, complemented, Sylow
100#534—1071no—
101#448—1072no—
102#393—1071no—
103#327—1072no—
104#323D10711yescharacteristic, complemented
105C22×Q8—1071no—
106C22×D4—2141no—
107C2×SD16—2142no—
108#123—2142no—
109#121—1072no—
110#121—2141no—
111#120—2142no—
112C2×C22⋊C4—2141no—
113C2×C22⋊C4C2×D10711yescharacteristic
114#103—2142no—
115C22⋊C8C2×D10712yescomplemented
116#95—1071no—
117#95—2141no—
118C24—2141no—
119C24D42811yescharacteristic
120C2×Q8—4282no—
121C2×Q8—2142no—
122C2×D4—1072no—
123C2×D4—4282no—
124C2×D4—4282no—
125C2×D4—2141no—
126C2×D4—2142no—
127C2×D4—2142no—
128C22×C4—2141no—
129C22×C4—2141no—
130C22×C4—2141no—
131C22×C4D42811yescharacteristic
132C22×C4C22×D10711yescharacteristic
133SD16—4284no—
134C2×C8—22no—
135C4⋊C4—1072no—
136C4⋊C4—2141no—
137C22⋊C4—22no—
138C22⋊C4—4282no—
139C22⋊C4—2142no—
140C22⋊C4—2142no—
141C23—21no—
142C23—21no—
143C23—41no—
144C23—41no—
145C23—8561no—
146C23—8561no—
147C23—4282no—
148C23—2142no—
149C23#1313380712yescomplemented
150C23C2×D42811yescharacteristic
151Q8—4282no—
152Q8—2144no—
153D4—8561no—
154D4—4282no—
155D4—4284no—
156D4—4282no—
157D4—4282no—
158C2×C4—21no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—4281no—
162C2×C4—4281no—
163C2×C4—4281no—
164C2×C4—4282no—
165C2×C4—4281no—
166C2×C4—4281no—
167C2×C4—2142no—
168C2×C4—2142no—
169C2×C4D4×D10712yes—
170C8—42no—
171C22—21no—
172C22—21no—
173C22—22no—
174C22—42no—
175C22—41no—
176C22—41no—
177C22—81no—
178C22—8564no—
179C22—4282no—
180C22—4282no—
181C22—4282no—
182C22#1681207911yescharacteristic
183C22C2×#1313380711yescharacteristic
184C22#1681213711yescharacteristic
185C4—22no—
186C4—41no—
187C4—8561no—
188C4—4282no—
189C4—4282no—
190C2—21no—
191C2—21no—
192C2—42no—
193C2—8562no—
194C2#2029278011yescharacteristic
195C2#2029280711yescharacteristic
196C2#2029281511yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#323×C107C2
socleC22×C107#120
prime radical#20293189C2
hypercenter#323D107
hypocenterC107#1601
LCSG ▷ C22×C4×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #20292768C2semidirect
3. #20292539C2semidirect
4. #20292362C2semidirect
5. #323×C107C2semidirect
6. C22×Dic214C22semidirect
16. C107⋊#95C22semidirect
18. C22⋊C8×C107C22semidirect
46. C23×C107SD16semidirect
99. C107#1601semidirect
104. #323D107semidirect
115. C22⋊C8C2×D107semidirect
149. C23#13133807semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202931892
#202927682
#202925392
#202923622
#323×C1072
#1601107